國立虎尾科技大學 |

Measures of symmetry for convex sets and stability

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Measures of symmetry for convex sets and stability/ by Gabor Toth.
Author:	Toth, Gabor.
Published:	Cham :Imprint: Springer, : 2015.,
Description:	xii, 278 p. :ill. (some col.), digital ; : 24 cm.;
Contained By:	Springer eBooks
Subject:	Convex and Discrete Geometry. -
Online resource:	http://dx.doi.org/10.1007/978-3-319-23733-6
ISBN:	9783319237336

Measures of symmetry for convex sets and stability
Toth, Gabor.

Measures of symmetry for convex sets and stability[electronic resource] /by Gabor Toth. - Cham :Imprint: Springer,2015. - xii, 278 p. :ill. (some col.), digital ;24 cm. - Universitext,0172-5939. - Universitext..

First Things First on Convex Sets -- Affine Diameters and the Critical Set -- Measures of Stability and Symmetry -- Mean Minkowski Measures.

This textbook treats two important and related matters in convex geometry: the quantification of symmetry of a convex set -- measures of symmetry -- and the degree to which convex sets that nearly minimize such measures of symmetry are themselves nearly symmetric -- the phenomenon of stability. By gathering the subject's core ideas and highlights around Grunbaum's general notion of measure of symmetry, it paints a coherent picture of the subject, and guides the reader from the basics to the state-of-the-art. The exposition takes various paths to results in order to develop the reader's grasp of the unity of ideas, while interspersed remarks enrich the material with a behind-the-scenes view of corollaries and logical connections, alternative proofs, and allied results from the literature. Numerous illustrations elucidate definitions and key constructions, and over 70 exercises -- with hints and references for the more difficult ones -- test and sharpen the reader's comprehension. The presentation includes: a basic course covering foundational notions in convex geometry, the three pillars of the combinatorial theory (the theorems of Caratheodory, Radon, and Helly), critical sets and Minkowski measure, the Minkowski-Radon inequality, and, to illustrate the general theory, a study of convex bodies of constant width; two proofs of F. John's ellipsoid theorem; a treatment of the stability of Minkowski measure, the Banach-Mazur metric, and Groemer's stability estimate for the Brunn-Minkowski inequality; important specializations of Grunbaum's abstract measure of symmetry, such as Winternitz measure, the Rogers-Shepard volume ratio, and Guo's Lp -Minkowski measure; a construction by the author of a new sequence of measures of symmetry, the kth mean Minkowski measure; and lastly, an intriguing application to the moduli space of certain distinguished maps from a Riemannian homogeneous space to spheres -- illustrating the broad mathematical relevance of the book's subject.

ISBN: 9783319237336

Standard No.: 10.1007/978-3-319-23733-6doiSubjects--Topical Terms:

672138
Convex and Discrete Geometry.

LC Class. No.: QA639.5

Dewey Class. No.: 516.1

Measures of symmetry for convex sets and stability
LDR:03179nam a2200349 a 4500 001 838646
003 DE-He213
005 20160415134451.0
006 m d
007 cr nn 008maaau
008 160616s2015 gw s 0 eng d
020 $a 9783319237336 $q (electronic bk.)
020 $a 9783319237329 $q (paper)
024 7 $a 10.1007/978-3-319-23733-6 $2 doi
035 $a 978-3-319-23733-6
040 $a GP $c GP
041 0 $a eng
050 4 $a QA639.5
072 7 $a PBMW $2 bicssc
072 7 $a PBD $2 bicssc
072 7 $a MAT012020 $2 bisacsh
072 7 $a MAT008000 $2 bisacsh
082 0 4 $a 516.1 $2 23
090 $a QA639.5 $b .T717 2015
100 1 $a Toth, Gabor. $3 1069914
245 1 0 $a Measures of symmetry for convex sets and stability $h [electronic resource] / $c by Gabor Toth.
260 $a Cham : $c 2015. $b Imprint: Springer, $b Springer International Publishing :
300 $a xii, 278 p. : $b ill. (some col.), digital ; $c 24 cm.
490 1 $a Universitext, $x 0172-5939
505 0 $a First Things First on Convex Sets -- Affine Diameters and the Critical Set -- Measures of Stability and Symmetry -- Mean Minkowski Measures.
520 $a This textbook treats two important and related matters in convex geometry: the quantification of symmetry of a convex set -- measures of symmetry -- and the degree to which convex sets that nearly minimize such measures of symmetry are themselves nearly symmetric -- the phenomenon of stability. By gathering the subject's core ideas and highlights around Grunbaum's general notion of measure of symmetry, it paints a coherent picture of the subject, and guides the reader from the basics to the state-of-the-art. The exposition takes various paths to results in order to develop the reader's grasp of the unity of ideas, while interspersed remarks enrich the material with a behind-the-scenes view of corollaries and logical connections, alternative proofs, and allied results from the literature. Numerous illustrations elucidate definitions and key constructions, and over 70 exercises -- with hints and references for the more difficult ones -- test and sharpen the reader's comprehension. The presentation includes: a basic course covering foundational notions in convex geometry, the three pillars of the combinatorial theory (the theorems of Caratheodory, Radon, and Helly), critical sets and Minkowski measure, the Minkowski-Radon inequality, and, to illustrate the general theory, a study of convex bodies of constant width; two proofs of F. John's ellipsoid theorem; a treatment of the stability of Minkowski measure, the Banach-Mazur metric, and Groemer's stability estimate for the Brunn-Minkowski inequality; important specializations of Grunbaum's abstract measure of symmetry, such as Winternitz measure, the Rogers-Shepard volume ratio, and Guo's Lp -Minkowski measure; a construction by the author of a new sequence of measures of symmetry, the kth mean Minkowski measure; and lastly, an intriguing application to the moduli space of certain distinguished maps from a Riemannian homogeneous space to spheres -- illustrating the broad mathematical relevance of the book's subject.
650 2 4 $a Convex and Discrete Geometry. $3 672138
650 0 $a Discrete geometry. $3 672137
650 0 $a Convex geometry. $3 685708
650 0 $a Mathematics. $3 527692
710 2 $a SpringerLink (Online service) $3 593884
773 0 $t Springer eBooks
830 0 $a Universitext. $3 881573
856 4 0 $u http://dx.doi.org/10.1007/978-3-319-23733-6
950 $a Mathematics and Statistics (Springer-11649)